Another Veech Triangle

We show that the triangle with angles Pi/12, Pi/3 and 7*Pi/12 has the lattice property and compute this triangle's Veech group.Comment: covers material in my 2006 PhD thesis, 9 pages, 5 figure

Immersions and translation structures I: The space of structures on the pointed disk

We define a moduli space of translation structures on the open topological disk with a basepoint and endow it with a locally-compact metrizable topology. We call this the immersive topology, because it is defined using the concept of immersions: continuous maps between subsets of translation surfaces that respect the basepoints and the translation structures. Immersions induce a partial ordering on the moduli space, and we prove the ordering is nearly a complete lattice in the sense of order theory: The space is only missing a minimal element. Subsequent articles will uncover more structure and develop a topology on the space of all translation structures.Comment: 34 pages, 4 figures. New appendix providing a comparison to McMullen's geometric topolog

Grid graphs and lattice surfaces

First, we apply Thurston's construction of pseudo-Anosov homeomorphisms to grid graphs and obtain translation surfaces whose Veech groups are commensurable to $(m,n,\infty)$ triangle groups. These surfaces were first discovered by Bouw and M\"oller, however our treatment of the surfaces differs. We construct these surfaces by gluing together polygons in two ways. We use these elementary descriptions to compute the Veech groups, resolve primitivity questions, and describe the surfaces algebraically. Second, we show that some $(m,n, \infty)$ triangle groups can not arise as Veech groups. This generalizes work of Hubert and Schmidt.Comment: This version includes a more detailed description of how to derive algebraic formulas for the surfaces using the Schwarz-Christoffel Mapping Theorem. There were other minor improvements as well. 29 pages, 8 figure

Truchet Tilings and Renormalization

The Truchet tiles are a pair of square tiles decorated by arcs. When the tiles are pieced together to form a Truchet tiling, these arcs join to form a family of simple curves in the plane. We consider a family of probability measures on the space of Truchet tilings. Renormalization methods are used to investigate the probability that a curve in a Truchet tiling is closed.Comment: 34 pages, 11 figures. New version mentions the results of the paper: Gabor Pete, Corner percolation on Z^2 and the square root of 17. Annals of Probability 36 (2008), No. 5, 1711-1747. http://projecteuclid.org/euclid.aop/122113876

Immersions and the space of all translation structures

A translation structure on a surface is an atlas of charts to the plane so that the transition functions are translations. We allow our surfaces to be non-compact and infinite genus. We endow the space of all pointed surfaces equipped with a translation structure with a topology, which we call the immersive topology because it is related to the manner in which disks can be immersed into such a surface. We prove that a number of operations typically done to translation surfaces are continuous with respect to the topology. We show that the topology is Hausdorff, and that the collection of surfaces with a fixed lower bound on the injectivity radius at the basepoint is compact.Comment: 31 pages, 1 figure. Comments welcom

An infinite surface with the lattice property II: Dynamics of pseudo-Anosovs

We study the behavior of hyperbolic affine automorphisms of a translation surface which is infinite in area and genus that is obtained as a limit of surfaces built from regular polygons studied by Veech. We find that hyperbolic affine automorphisms are not recurrent and yet their action restricted to cylinders satisfies a mixing-type formula with polynomial decay. Then we consider the extent to which the action of these hyperbolic affine automorphisms satisfy Thurston's definition of a pseudo-Anosov homeomorphism. In particular we study the action of these automorphisms on simple closed curves and on homology classes. These objects are exponentially attracted by the expanding and contracting foliations but exhibit polynomial decay. We are able to work out exact asymptotics of these limiting quantities because of special integral formula for algebraic intersection number which is attuned to the geometry of the surface and its deformations.Comment: minor revisions, 29 pages, 7 figure

An Infinite Surface With The Lattice Property I: Veech Groups and Coding Geodesics

We study the symmetries and geodesics of an infinite translation surface which arises as a limit of translation surfaces built from regular polygons, studied by Veech. We find the affine symmetry group of this infinite translation surface, and we show that this surface admits a deformation into other surfaces with topologically equivalent affine symmetries. The geodesics on these new surfaces are combinatorially the same as the geodesics on the original.Comment: 23 pages, 10 figures. Improved exposition. Theorem 10 is an improved geodesic coding result. arXiv admin note: text overlap with arXiv:0802.018

Rel leaves of the Arnoux-Yoccoz surfaces

We analyze the rel leaves of the Arnoux-Yoccoz translation surfaces. We show that for any genus $g \geq 3$, the leaf is dense in the connected component of the stratum $H(g -1 , g -1)$ to which it belongs, and the one-sided imaginary-rel trajectory of the surface is divergent. For one surface on this trajectory, namely the Arnoux-Yoccoz surface itself, the horizontal foliation is invariant under a pseudo-Anosov map (and in particular is uniquely ergodic), but for all other surfaces, the horizontal foliation is completely periodic. The appendix proves a field theoretic result needed for denseness of the leaf: for any $n \geq 3$, the field extension of the rationals obtained by adjoining a root of $X^n-X^{n-1}-\ldots-X-1$ has no totally real subfields other than the rationals.Comment: Appendix by Lior Bary-Soroker, Mark Shusterman and Umberto Zannier. The prior version was published, but had errors in \S 6. Erroneous statements have been indicated and an erratum was included as Appendix B which corrects the errors. Main results are still correct. 70 pages, 9 figures. arXiv admin note: text overlap with arXiv:1506.0677

The rel leaf and real-rel ray of the Arnoux-Yoccoz surface in genus 3

We analyze the rel leaf of the Arnoux-Yoccoz translation surface in genus 3. We show that the leaf is dense in the stratum $\mathcal{H}(2,2)^{\mathrm{odd}}$ but that the real-rel trajectory of the surface is divergent. On this real-rel trajectory, the vertical foliation of one surface is invariant under a pseudo-Anosov map (and in particular is uniquely ergodic), but the vertical foliations on all other surfaces are completely periodic

Indiscriminate covers of infinite translation surfaces are innocent, not devious

We consider the interaction between passing to finite covers and ergodic properties of the straight-line flow on finite area translation surfaces with infinite topological type. Infinite type provides for a rich family of degree $d$ covers for any integer $d>1$. We give examples which demonstrate that passing to a finite cover can destroy ergodicity, but we also provide evidence that this phenomenon is rare. We define a natural notion of a random degree $d$ cover and show that, in many cases, ergodicity and unique ergodicity are preserved under passing to random covers. This work provides a new context for exploring the relationship between recurrence of the Teichm\"uller flow and ergodic properties of the straight-line flow.Comment: Improved exposition thanks to referee's comments. 54 pages, 9 figure